A QCD-Analysis for Radiative Decays of $\Upsilon$ into $f_2(1270)$

We perform a QCD analysis for the radiative decay of a heavy $^3S_1$ quarkonium into the tensor meson $f_2(1270)$. We make an attempt to separate the nonperturbative effect related to the quarkonium and that related to the tensor meson, the former is represented by NRQCD matrix elements, while the later is parameterized by distribution amplitudes of gluons in the tensor meson at twist-2 level and at twist-3 level. We find that at twist-2 level the helicity $\lambda$ of the tensor meson can be 0 and 2 and the amplitude with $\lambda =2$ is suppressed. At twist-3 level the tensor meson can have $\lambda =1$. A comparison with experiment is made, an agreement of our results with experiment can be found. We also briefly discuss the radiative decay into $\eta$ and obtain a prediction for $\Upsilon\to\gamma+\eta$.